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Comprehensive quasi-Einstein spacetime with application to general relativity

The aim of this paper is to extend the notion of all known quasi-Einstein manifolds like generalized quasi-Einstein, mixed generalized quasi-Einstein manifold, pseudo generalized quasi-Einstein manifold and many more and name it comprehensive quasi Einstein manifold C(QE)$_{n}$. We investigate some geometric and physical properties of the comprehensive quasi Einstein manifolds C(QE)$_{n}$ under certain conditions. We study the conformal and conharmonic mappings between C(QE)$_{n}$ manifolds. Then we examine the C(QE)$_{n}$ with harmonic Weyl tensor. We investigate geometric and physical properties of the comprehensive quasi Einstein manifolds C(QE)$_{n}$ under certain conditions. We define the manifold of comprehensive quasi-constant curvature and proved that conformally flat C(QE)$_{n}$ is manifold of comprehensive quasi-constant curvature and vice versa. We study the general two viscous fluid spacetime C(QE)$_{4}$ and find out some important consequences about C(QE)$_{4}$. We study C(QE)$_{n}$ with vanishing space matter tensor. Finally, we prove the existence of such manifolds by constructing non-trivial example.